Condensed Matter > Statistical Mechanics

Title:
Quantifying the Rise and Fall of Complexity in Closed Systems: The Coffee Automaton

Abstract: In contrast to entropy, which increases monotonically, the "complexity" or
"interestingness" of closed systems seems intuitively to increase at first and
then decrease as equilibrium is approached. For example, our universe lacked
complex structures at the Big Bang and will also lack them after black holes
evaporate and particles are dispersed. This paper makes an initial attempt to
quantify this pattern. As a model system, we use a simple, two-dimensional
cellular automaton that simulates the mixing of two liquids ("coffee" and
"cream"). A plausible complexity measure is then the Kolmogorov complexity of a
coarse-grained approximation of the automaton's state, which we dub the
"apparent complexity." We study this complexity measure, and show analytically
that it never becomes large when the liquid particles are non-interacting. By
contrast, when the particles do interact, we give numerical evidence that the
complexity reaches a maximum comparable to the "coffee cup's" horizontal
dimension. We raise the problem of proving this behavior analytically.